The metric compatibility condition in the definition of Levi-Civita connection says that

g=ea⊗ea.
g = e^a \otimes e_a
\,.

The torsion-freeness condition says that

Fe=0.
F_e = 0
\,.

In terms of Christoffel symbols

The Levi-Civita connection may be discussed in terms of its components – called Christoffel symbols – given by the canonical local trivialization of the tangent bundle over a coordinate patch. This has been the historical route and is still widely used in the literature.

Metric compatibility

Here a metric gg is compatible with the connection ∇\nabla or preserved by it (here thought of in its incarnation as a covariant derivative) if and only if ∇Xg=0\nabla_X g = 0 for all XX, which is equivalent to the preservation of the metric inner product of tangent vectors under parallel translation. Since

Now assume M⊂ℝnM \subset \mathbb{R}^n and we have such a connection associated to gg.Then the connection is uniquely determined by its Christoffel symbols, which we can determine in terms of gg by a bit of elementary algebra. In other words, we just need to compute ∇∂i∂j\nabla_{\partial_i} \partial_j. Now

These are three linear equations in the unknowns g(Sik,∂j),g(Sjk,∂i),g(Sij,∂k)g( S_{i k}, \partial_j), g( S_{j k}, \partial_i), g( S_{i j}, \partial_k). The system is nonsingular, so we get a unique solution, and consequently by nondegeneracy a unique possibility for the SijS_{i j}.

Incidentally, we have in fact shown the uniqueness assertion of the general theorem, since that is local.

We shall now prove existence in this restricted case. Choose SijS_{i j} to satisfy the system of three equations outlined above where i<j<ki \lt j \lt k. Then set Sji:=SijS_{j i} := S_{i j}, and we have a connection ∇\nabla with ∇∂i∂j:=Sij\nabla_{\partial_i} \partial_j := S_{i j} since the vector fields ∂i\partial_i are a frame (i.e. a basis at each tangent space on MM). It is symmetric, since the torsion TT vanishes (by Sij=SjiS_{i j}=S_{j i}) on pairs (∂i,∂j)(\partial_i,\partial_j), and hence identically, since it is a tensor.

We must check for compatibility. The difference of the two terms in (1) vanishes when X,X1,X2X,X_1,X_2 are of the form ∂i\partial_i. The vanishing holds generally because the difference of the two sides, which is (∇Xg)(X1,X2)(\nabla_X g)(X_1,X_2), is a tensor. Hence compatibility follows.

Uniqueness and existence in the general case

We have already shown the uniqueness assertion, since that is local. Connections restrict to connections on open subsets.

We have proved the existence of ∇\nabla when MM is an open submanifold of ℝn\mathbb{R}^n (though not necessarily with the canonical metric ∑i=1ndxi⊗dxi\sum_{i=1}^n d x_i \otimes d x_i). In general, cover MM by open subsets UiU_i diffeomorphic to an open set in ℝn\mathbb{R}^n. We get connections ∇i\nabla_i on UiU_i compatible with g|Uig|_{U_i}.

We claim that ∇i|Ui∩Uj=∇j|Ui∩Uj\nabla_i|_{U_i \cap U_j} = \nabla_j|_{U_i \cap U_j}. This is an easy corollary of uniquness. So we can patch the connections together to get the one Levi-Civita connection on MM.